Let's solve the system of equations step by step to find their solution:
The given system of equations is:
[tex]\[
\begin{cases}
y = -x + 3 \\
y = -2x - 1
\end{cases}
\][/tex]
To find the solution, we need to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy both equations. Since both expressions are equal to [tex]\( y \)[/tex], we can set the right-hand sides of the equations equal to each other:
[tex]\[
-x + 3 = -2x - 1
\][/tex]
Now, isolate [tex]\( x \)[/tex] by combining like terms:
1. Add [tex]\( 2x \)[/tex] to both sides of the equation:
[tex]\[
x + 3 = -1
\][/tex]
2. Next, subtract 3 from both sides to isolate [tex]\( x \)[/tex]:
[tex]\[
x = -4
\][/tex]
Now that we have the value of [tex]\( x \)[/tex], we can substitute it back into either of the original equations to find [tex]\( y \)[/tex]. Let's use the first equation [tex]\( y = -x + 3 \)[/tex]:
[tex]\[
y = -(-4) + 3
\][/tex]
[tex]\[
y = 4 + 3
\][/tex]
[tex]\[
y = 7
\][/tex]
Therefore, the solution to the system of equations is [tex]\( (x, y) = (-4, 7) \)[/tex].
To graph these equations, you would plot the two lines given by the equations [tex]\( y = -x + 3 \)[/tex] and [tex]\( y = -2x - 1 \)[/tex]. The point of intersection of these lines represents the solution to the system of equations. Thus, the graph would show the lines intersecting at the point [tex]\( (-4, 7) \)[/tex].
In conclusion, the solution to the system of equations [tex]\( y = -x + 3 \)[/tex] and [tex]\( y = -2x - 1 \)[/tex] is [tex]\( (x, y) = (-4, 7) \)[/tex], and the graphs of these equations intersect at this point.
